Friday, March 14, 2014

Calculus: e and pi Are Both Transcendental

Yesterday I was introducing the number e, and telling the students that my preferred definition of e is "the number that makes a slope of 1 for the graph of y=ax at x=0." I also told them that this definition makes mathematicians unhappy, and wrote out the limit definition. But I like them seeing a concrete meaning for this strange number.

As I made this introduction, I mentioned that, like pi, e is irrational (not a ratio of whole numbers) and transcendental (not the solution to an algebra equation using whole numbers). I showed them the proof that the square root of 2 is irrational. But it is the solution to x2=2, so it's an algebraic number, not a transcendental number. I told them that I have not yet completely understood the proof that pi is irrational. (Updated link: Brent Yorgey, one of my favorite bloggers, posted a good series explaining it at The Math Less Traveled. But I never managed to get all the way through it.)  "So it's not really math for me to say pi and e are irrational, when I don't know it by a proof but only by believing what I've been told."

And then I decided that since pi was in the air, and Pi Day was coming, we'd do an activity I'll be doing today with a different group. I had them all stand up around the outer edges of the room, and hold hands, arms outstretched. The extra people made a diameter along the middle of the room. While I was getting them to stretch, I pulled people out by saying "You're out." They laughed. I told them it was a good kind of out, that we were the circle makers.

The reason I'm posting this silly, super simple activity? When we were done, and they counted off, it turned out that there were 22 around and ... (you know I was holding my breath) ...  7 across the middle! I couldn't believe it. I was glowing, further proof to my students that I am a total nerd for math.  ;^)

 In case you're wondering why getting 22 around and 7 across was so special, that makes circumference / diameter = 22/7 = 3.1428, almost perfect. And before calculators made people think decimals were cooler than fractions, 22/7 was the estimate for pi.

(Added on 3/14: Tried it again today in a smaller space. We got 17/4. Bummer.)


  1. I'm honored to find out I am one of your favorite bloggers! =) You can find the proof that pi is irrational here: Scroll down to "Irrationality of Pi". The proof that e is irrational is actually much simpler than the proof for pi --- I hope to write about it at some point, once I get back to blogging regularly again (at the moment I'm trying to finish my PhD dissertation). It seems there are proofs of the transcendence of e and pi which are about the same level of difficulty as the proof of pi's irrationality, but I haven't looked carefully at them myself.


  3. Hi Sue,

    you should check out Michael Spivak's Calculus. The entire 16th chapter is devoted to proving the irrationality of Pi. He's very thorough and doesn't skip "obvious" steps like Niven's classic half-page proof linked above (although Niven's proof is not aimed at the same audience, e.g. professional mathematicians). Spivak's (treatment of the) proof was the first proof that I was able to grasp.

    Also, you should also see this (other classic) integral @ (it's in french but you'll get the math parts for sure) which links nicely 22/7 and Pi.

    Cheers !


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